Abstract
Given a field π of characteristic 2 and an integer n β₯ 2, let W(2n β 1, π) be the symplectic polar space defined in PG(2n β 1, π) by a non-degenerate alternating form of V(2n, π) and let Q(2n, π) be the quadric of PG(2n, π) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n β 1, π) β Q(2n, π). This is true when π is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n β 1, π) is indeed isomorphic to a non-singular quadric Q, but when π is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, π) is a proper subgeometry of W(2n β 1, π). We show that, in spite of this fact, W(2n β 1, π) can be embedded in Q(2n, π) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n β 1, π) of W(2n β 1, π) into the dual DQ(2n, π) of Q(2n, π).